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Negative curves on very general blow-ups of P^2

Tommaso de Fernex
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Abstract

A conjecture, related to the Nagata conjecture and the Segre-Harbourne-Gimigliano-Hirschowitz conjecture, states that every integral curve with negative self-intersection on the blow-up of $¶^2$ at a set of points in very general position is a (-1)-curve. In this paper we verify this conjecture for all curves whose image on $¶^2$ is a curve with a singularity of multiplicity two at one of the centers of the blow-up.

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Tommaso de Fernex. 2005-12-29. Negative curves on very general blow-ups of P^2. https://arxiv.org/abs/math/0512631

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